Spectrally shaped transient forcing functions for frequency response testing

Journal of Sound and Vibration

(1972) 23 (3), 307-3 18

SPECTRALLY SHAPED TRANSIENT FORCING FUNCTIONS FOR FREQUENCY RESPONSE TESTING R. G. WHITE Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 SNH, England (Received 9 March 1972)

The transfer characteristics of dynamic force generating systems for frequency response measurements on mechanical systems often cause the generated force spectrum to differ in shape from the input voltage spectrum. By considering the spectral characteristics of the linear rapid frequency sweep, it is shown that good estimates of the modulus spectra of swept sinewaves with varying sweep rates may he calculated. It is shown how swept sinewave voltages of constant amplitude may be generated with an appropriately predetermined sweep rate to allow the transfer characteristics of the force generating system to be compensated for, thus producing a spectrally shaped transient forcing function at the output of the force generator. The application of the technique in vibration testing is demonstrated.

1. INTRODUCTION Considerable interest has been shown recently in the use of transient methods of measuring the frequency response of physical systems. This is largely due to the development of digital computer-based data analysis systems with fast Fourier transform routines. The particular application of such techniques for the measurement of the frequency response characteristics of structures has been studied in depth [l, 2, 31, various aspects of transient excitation and analysis techniques being examined. Later work [4] has shown that mechanical impedance measurements may be made very rapidly on structures by using transient excitation and a digital data analysis system, the accuracy achieved comparing very well with that attained by more traditional methods. The approach to the problem of experimental measurement involves consideration of three basic requirements : to generate a suitable forcing function, to measure the applied force and the response of the test system, and to analyse the test data. For structural studies, the forcing function could be generated mechanically and transient tests have been successfully carried out in which such a system was used [5]. However, the more conventional approach is to generate the required voltage waveform and excite the structure by using a power amplifier and electrodynamic exciter. Force and response measurement is most often carried out by using piezoelectric transducers with associated signal processing circuits and amplifiers. To be meaningful to the analyst, the derived frequency response data should be presented in the conventional form of modulus and phase spectra or in vector form, this being achieved from the time data by Fourier transformation. The Fourier transform of a signal may be evaluated directly to yield the complex spectrum or the intermediate stage of correlation may be introduced in order to reject extraneous noise [3] present in the signal. Consideration of a variety of transient forcing functions which could be electronically generated and are suitable for use in frequency response measurement [ 1] has shown that the linear swept sinewave is very suitable because its modulus spectrum is essentially rectangular in shape. This gives a high degree of control over the range of resonances excited and the use 307

308

R. G. WHITE

of correlation techniques, already mentioned above, is facilitated because of the flatness of the excitation spectrum. It is often apparent, however, in frequency response measurement experiments on mechanical systems, that although the required forcing function may be precisely generated electronically, nevertheless, because of the characteristics of the force generating system, the resulting applied excitation spectrum differs considerably from that desired. This is because the transfer function of the instrumentation between the voltage generator and the excitation point modifies the spectral characteristics of the input signal. This effect is particularly evident if wide frequency ranges are to be investigated by using electrodynamic exciters or vibration tables. This paper shows how the rapid frequency sweep waveform may be used to generate spectrally shaped transient forcing functions. This permits compensation for the transfer function of the excitation system so that a transient force with a flat modulus spectrum may be applied to the system under test. 2. THE LINEAR FREQUENCY SWEEP The frequency sweep, or swept sinewave, is defined by O
F(t) = FOsin 4(t),

(1)

The function is therefore of constant amplitude and time-dependent frequency, the instantaneous variation of frequency with time being given by w(t) =

d+(t)

(2)

dt.

The use of the swept sinewave in structural frequency response measurement has been examined [l, 6, 71 and the linear frequency sweep has been shown to be most suitable as a transient forcing function because the modulus spectrum is “rectangular”. For frequency ranges of approximately one decade, sweep times, T, of the order of one second have been shown to be sufficient to allow enough energy to pass into the system to produce measurable response levels. A linear variation of frequency with time is given by F(t) = F. sin (at2 + bt)

(3)

where o(t) = 2at + b and dw(t)/dt = 2a, with a = (w2 - w,)/2Tand b = w,, w, and w2 being the initial and final frequencies. To examine the spectral characteristics of the function defined in equation (3), it is necessary to evaluate the Fourier transform, defined by P(iw) = 1 F(t) eeiW’dt. --m

(4)

The Fourier transform of the linear frequency sweep is evaluated in Appendix A and the modulus is given by equation (A7) as

(5) where A

=

4x1)

-

f(X2),

B

=

4x*)

-

4x2)s

c

=

s(x3)

-

44

D

=

c(x3)

-

c(xq).

309

SPECTRALLY SHAPED TRANSIENTS s(x)

and c(x) are the sine and cosine Fresnel integrals. Also

The presence of the Fresnel terms in equation (5) precludes further reduction to give a simple exact expression for the modulus spectrum. However, the spectrum characteristics may be examined in the following manner. The functions, x, in the various frequency ranges are shown below in Table 1. TABLE

I

The frequency functions

w=o

Heredw=w,-w,ando,=----.

w = WI

WI

w1
w =

w2

w * WI

+w 2

In the region w, < w < w2, x1, x2 and x3 are positive and x4 is negative. If w is not close to either of the cut-off frequencies, then the mean value about which s(x) and c(x) oscillate is 0.5. By using conventional notation to indicate mean values, and noting that s(- 1x41)= -4%)

and

c(- 1x41)=-4x4),

it may be shown that ;?z=$=O and

c2=@=1_

It may therefore be shown that the mean modulus spectrum level from equation (5) in the above defined frequency range is ~(w)j=~,&.

(6)

The modulus spectrum levels at the initial and final frequencies, wi and w2, may be examined in a similar manner. It may be seen from Table 1 that at w,, x1, x2 and x3 are positive and x4 = 0, and that at w2, x1 and x2 are positive, x3 = 0 and x4 is negative. Now, carrying out an approximate analysis, with the mean values as above being used, yields, for the spectrum levels at w, and w2,

310

R. G.

WHITE

Thus the modulus spectrum is symmetrical and comparison of equations (6) and (7) shows that the modulus spectrum level at w1 and w2 is half the mean level of the spectrum in the central frequency region. It may also be seen by similar analyses that

In order to demonstrate the above characteristics, a rapid frequency sweep defined by equation (3) was generated by digital computer and its Fourier transform evaluated. The modulus and phase spectra of the Fourier transform of a sweep from 20 to 220 Hz in 1 s are shown in Figure 1, and the flatness of the modulus spectrum between the peaks at the extremities is apparent. Linear scales are used for modulus spectrum plots throughout this work. The predicted mean spectrum level and the levels atfr andf2 are also shown in the same figure. It can be seen that there is good agreement between the predicted and measured characteristics. The modulus spectrum is plotted in units of volts because digital data analysis

En,, , , , , , , , , j2u 0

20 40

60

Bo

100Ix)140I601602C0220%lO260280300

Frequency (Hz)

Figure 1. The Fourier transform of a linear frequencysweep. systems generate a discrete Fourier transform from a finite set of data samples but the equivalence of the discrete and integral transforms has been established [9]. The result of the approximate analysis to give the mean spectrum level according to equation (6) agrees with the results of other workers [6,7] who used much lower sweep rates. Thus, in rapid frequency sweep tests, the excitation level may be predicted. It must be noted however that if the sweep rate is extremely high, for example if a wide frequency range is swept in milliseconds, then there is experimental evidence [lo] to show that the modulus spectrum becomes “smoothed” and does not exhibit the rectangular characteristic. However, the extremely high sweep rates used in the work reported in reference [IO] would probably not be suitable for structural testing because of response signal to noise ratio considerations. 3. VARYING SWEEP RATES The spectral characteristics of a swept sinewave have been examined in the previous section and such a forcing function could be applied to a system in a frequency response test. The effect of the transfer function between the voltage generator and the point of application of the force to a system has already been discussed in the introduction. It is often difficult to

SPECTRALLY

311

SHAPED TRANSIENTS

preserve the spectral characteristics of the forcing function in a practical test; there is, therefore, a need to generate a waveform which when passed through the excitation system will produce the desired flat modulus spectrum at the output of the complete force generating system. This is often accomplished in steady-state sinusoidal or random excitation testing by using a spectrum shaper consisting of many narrow-band filters to “weight” the input voltage spectrum. Equation (6) may be rewritten to show that the modulus spectrum level of the linear rapid frequency sweep is

where c;,is the rate of change of frequency with time. Thus, if a given bandwidth dw is swept, the spectrum level within the bandwidth is controlled by F,, and the time taken to sweep through the bandwidth. The r.m.s. force level could be controlled by using filters as described above but the introduction of reactive components, particularly if narrow band filters are necessitated, could cause a further problem due to “ringing” occurring in transient testing. The dynamic force range could also become large in some instances. Both problems may be obviated by maintaining 8” constant and varying the sweep rate to control the modulus spectrum level. The analyses already described in section 2 have shown that the Fourier transform of a swept sinewave with a constant sweep rate is a complicated function of frequency involving Fresnel integrals. Analytical evaluation of equation (4) would be impossible for more complicated functions, F(t); hence exact calculation of the spectral characteristics of functions with varying sweep rates is not generally possible directly from the Fourier transform. However, if the instantaneous sweep rate at any frequency is known, then equation (9) may be used to predict the modulus spectrum level at that frequency. This approach is examined below. Several frequency sweep functions were considered in order to illustrate the principle of spectrum shaping via sweep rate control. Two functions are presented here and denoted according to the variation of frequency with time. (i) Square law sweep : F(t) = F. sin (ct3 + bt), w(t) = 3ct2 + b, b-w,,

O
c = (co2 - w,)/3T2;

(10)

(ii) Sine law sweep : F(t)=F,sin[k(cosg-I)+bt], w(t) = - ETsin 5: + b,

b=w,,

2T

k = -+I+

O
- co,).

2T’

(11)

Now, unlike the linear frequency sweep, for which d = 2a from equation (3), the sweep rates above are a function of time. For the square law sweep, C is a minimum at w1 and for the sine law sweep the rate is a minimum at w2. Therefore, according to equation (9), the maximum modulus spectrum levels are predicted at w1 and w2, respectively, for the two functions. 21

312

R. G. WHITE

The two functions were generated by digital computer for the same frequency range and sweep time used previously in the linear frequency sweep 20 to 220 Hz in 1 s. The waveforms are shown in Figures 2(a) and (c). Modulus spectra of the Fourier transforms are given in Figures 2(b) and(d) and the overall characteristics are as deduced above. Equation (9) in conjunction with the appropriate relationships for w(t) and h(t) was used to calculate the modulus spectrum levels given by the dashed curves in Figure 2(b). Very close agreement between predicted and measured spectra is apparent. This was further confirmed by applying the same procedure to a range of swept sinewaves with varying sweep rates, such as exponential and cubic variation of frequency with time; good agreement between predicted and measured modulus spectrum characteristics was obtained in each case.

i

Time b)

Time Cd)

Frequency (Hz)

Frequency

(Hz)

Figure 2. Waveforms and Fourier transforms of swept sinewaves with varying sweep rates. (a) Square law frequency sweep waveform. (b) Fourier transform of the square law frequency sweep. (c) Sine law frequency sweep waveform. (d) Fourier transform of the sine law frequency sweep. (b) and (d): ---, caiculated level.

4. A PRACTICAL

SYSTEM

It has been shown that spectrally shaped transient forcing functions may be generated by using constant amplitude swept sinewaves with varying sweep rates. The functions used were generated digitally; it therefore remains to demonstrate the practical application of the technique using electronically generated waveforms. An established technique for generating rapid frequency sweeps is to use a voltage tuned oscillator driven by a voltage generator. Such an oscillator was used for the frequency response measurements reported in references [l-4] and the design of the instrument is fully detailed in reference [l 11. For a linear frequency sweep the driving voltage is ramp from the initial voltage to the final voltage in the sweep time T.Variation in sweep rate may obviously be achieved by shaping the driving voltage to the oscillator. This method is used here in the demonstration of the experimental technique. The apparatus used is shown in Figure 3. The oscillator was used to drive an electrodynamic exciter through a power amplifier. The response of a mass mounted on the exciter wasmeasured

313

SPECTRALLY SHAPED TRANSIENTS

by a piezoelectric accelerometer. After amplification, the acceleration signal was input to a two-channel, analogue to digital converter coupled to a digital computer, the computer having been programmed for data analysis. The other channel of the analogue to digital converter was connected directly to the output of the sweep oscillator. A linear frequency sweep from 0.1 to 1 kHz in 1 s was generated and applied to the power amplifier. The oscillator output voltage spectrum is shown in Figure 4(a), and the accelerometer signal spectrum is shown in Figure 4(b). Examination of Figure 4(b) shows the excitation system to exhibit gain in the low frequency region and “cut-off” in the high frequency region. This is made very clear in the transfer function presented in Figure 5. This was obtained by the complex division of the Fourier transforms of the oscillator output signal and the accelerometer signal; the phase information is presented in Figure 5 for completeness. It is worthy of note that although this particular transient test technique has been developed for vibration analyses, it does offer a rapid method of instrument calibration, the data in Figures 4 and 5 being computed and plotted in a few minutes.

Cccelerometer

Digitalcomputer

Figure 3. Block diagram of instrumentation.

To control the sweep rate, a diode function generator was inserted between the ramp generator and the oscillator. The diode function generator used, Philbrick Nexus SPFX-P, gave an eleven-straight-line respresentation of the required function with ten break-points. Input and output amplifiers to the function generator were used in order that the full 10 volt range of the device could be used. The device was connected such that non-monotonic functions could be generated. The required input function to the oscillator was calculated from Figure 4(b). Mean spectrum levels in constant bandwidths &-were measured from Figure 4(b) and tabulated. The ratio of the maximum level to the level in each bandwidth was then calculated. With equation (9) written as ~=#$?

(12)

the ratios of the sweep times required for each bandwidth were then derived from the level ratios calculated above, the total sweep time being 1 s as previously. The calculated driving voltage to be generated is shown plotted in Figure 6 and the achieved curve is shown plotted in the same figure. The output voltage spectrum from the oscillator is given in Figure 7(a) and shows how the high frequency content of the spectrum had been increased in order to compensate for the high frequency cut-off in the exciter system. The acceleration spectrum recorded from the mass mounted on the exciter is shown in Figure 7(b). It can be seen that the high frequency content of the spectrum had been enhanced and the spectrum is considerably flatter than that in Figure 4(b).

314

R. G. WHITE

Frequency (Hz)

Figure 4. Oscillator voltage and accelerometer signal spectra when a linear frequency sweep excitation was used. (a) Spectrum of oscillator output voltage. (b) Spectrum of acceleration signal.

I

0

I

200

I

I,

400

I, 600

, 800

,

I, ICC0

1200

Frequency (Hz) 2rr r

Figure 5. Transfer function between oscillator output and acceleration.

The fluctuations in the spectrum shown in Figure 7(b) are due to the fact that, with the diode function generator used, the slopes were set on single turn potentiometers. The degree of control would be considerably increased by using multi-turn potentiometers. The peak at the high frequency end of the spectrum was caused by the usually evident peak near the upper frequency limit (see, for instance, Figure 1 where this is clearly apparent) being increased by the very low sweep rate at the final frequency. This effect is also apparent in the spectrum of

315

SPECTRALLY SHAPED TRANSIENTS

0

I

0.I

I 0.2

I

Cl.3

I

0.4

II 0.5

0.6

I

0.7

I1

0.8

0.9

I

IO

Time (s)

Figure 6. The oscillator driving voltage. x, Calculated values of required function; -, at output from diode function generator.

voltage achieved

Frequency (Hz)

Figure 7. Oscillator voltage and accelerometer signal spectra when a varying sweep rate was used. (a) Spectrum of oscillator output voltage. (b) Spectrum of acceleration signal.

the sine law sweep in Figure 2. The sweep rate could not be increased in the region of the peak because of the limited number of breakpoints; 100 Hz bandwidths were used in the spectrum calculations above. The use of a diode function generator with many more, or variable, breakpoints would enable regions of rapid fluctuations in the spectrum to be controlled more accurately. 5. CONCLUSIONS

The spectral characteristics of the linear swept sinewave have been investigated. It has been shown that from these characteristics the modulus spectra of rapidly swept sinewaves with constant amplitude and varying sweep rates may be accurately estimated.

316

R. G. WHITE

The characteristics of vibration test systems may be compensated for in a transient test by controlling the parameters of the swept sine input voltage, thus enabling spectrally shaped transient forcing functions to be generated at the output of the force generating system. Particular application of the technique in vibration testing has been demonstrated and necessary instrumentation characteristics have been studied. The method, however, shows promise for frequency response measurement in other fields.

ACKNOWLEDGMENTS The author wishes to thank Mr G. C. Wright and Mr P. J. Holmes of the Institute of Sound and Vibration Research for their assistance in carrying out the various digital computations necessary in this work. REFERENCES 1.

2. 3. 4. 5. 6. 7. 8. 9.

10. 11.

R. G. WHITE 1968 Journal of the Royal Aeronautical Society 73, 1047-1050. Use of transient excitation in the dynamic analysis of structures. R. G. WHITE 1971 Journal of Sound and Vibration 15, 147-161. Evaluation of the dynamic characteristics of structures by transient testing. F. KANDIANIS1971 Journal of Sound and Vibration 15,203-215. Frequency response of structures and the effects of noise on its estimates from the transient response. P. J. HOLMES1972 Institute of Sound and Vibration Research Technical Report 53. Mechanical impedance measurement by the transient loading technique. A. J. FRANCKEN1969 Institute T.N.O. for Mechanical Construction, Delft. Mechanical admittance methods. W. H. REED, A. W. HALL and L. E. BARKER 1960 NASA TN 0508. Analogue techniques for measuring the frequency response of linear physical systems excited by frequency sweep inputs. G. P. THRALL, D. A. POPE and R. K. OTNES 1966 Measurement Analysis Corporation Report MAC 506-10. Studies of frequency response functions using swept sine inputs. M. ABRAMOWITZ and I. A. STEGUN 1965 Handbook of Mathematical Functions. New York: Dover Publications. J. W. COOLEY,P. A. W. LEWIS and P. D. WELCH 1969 Applications and methods of random data analysis. Lecture series ZSVR, University of Southampton. The fast Fourier transform algorithm: programming and accuracy considerations in the calculation of sine, cosine and Laplace transforms. P. T. BRADY, A. S. HOUSE and K. N. STEVENS1961 Journal of the Acoustical Society of America 33, 1357-1362. Perception of sounds characterised by a rapidly changing resonant frequency. R. G. WHITE 1969 Znstitute of Sound and Vibration Research Technical Report 12. Measurement of structural frequency response by transient excitation.

APPENDIX

I

THE FOURIERTRANSFORM OF THE LINEAR FREQUENCY SWEEP

The linear frequency

sweep is defined by F(t) = F. sin (at2 + bt),

O
where a = ((02 - w,)/2T, q

and w2 are the initial and final frequencies,

evaluated

b=w,. respectively.

is

F(iw) =

1F(t)

e-l”‘dt,

(Al) The Fourier

transform

to

be

SPECTRALLY

SHAPED

317

TRANSIENTS

which, because of the conditions of equation (Al), may be written as (A2)

F(iw) = i F(1) e-‘“‘dt. 0

Substituting (Al) into (A2) and resolving into real and imaginary components gives /sin(ot2+bt)coswrdt-i?sin(at”+bt)sinwrdt

zr-

Fo

I

0

0 T

T

T

sinXdt +

2 s

s

sin Ydt + i

s

cos Xdt - i

(A3)

0

0

0

where h=b-w.

g=b+o,

Y=at2+ht,

X=aP+gt,

Now the integrals in equation (A3) may be written [8] as

etc., where s [ ] and c[ ] are the sine and cosine Fresnel integrals. Therefore, from equation (A3), I;(iw) = $

J-

5 [efe2/4a{S(G)+ k(G)} + e-fh*/4a{S(H) - k(H)}]:,

(A4)

2 J+h2’4a{s(X3) -

(A%

where

Substituting the limits gives F(iw)

& [efg2/4~WI)

=

-

s(x2)

+

i&>

-

i&2))

4x4) - ic(x3)+ ic(x4)}],

+e

-

Resolving into real and imaginary components gives F(iw) =2

J ([ &

Acosgi-Bsini+

Ccoshi-

Dsing

I

2

Asing&+BcosgG-CsinhG-Dcosh& where A

=

+I)

-

dx2),

B

=

ch)

-

c&2),

c

=

~(~3)

-

&x4),

D = c(x3) - c(x4).

,

646)

318 The modulus spectrum is therefore

Substituting for g and h gives, finally,

R. G. WHITE